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Volatility of Volatility Estimation: Central Limit Theorems for the Fourier Transform Estimator and Empirical Study of the Daily Time Series Stylized Facts

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Listed:
  • Giacomo Toscano
  • Giulia Livieri
  • Maria Elvira
  • Stefano Marmi

Abstract

We study the asymptotic normality of two feasible estimators of the integrated volatility of volatility based on the Fourier methodology, which does not require the pre-estimation of the spot volatility. We show that the bias-corrected estimator reaches the optimal rate n1/4, while the estimator without bias-correction has a slower convergence rate and a smaller asymptotic variance. Additionally, we provide simulation results that support the theoretical asymptotic distribution of the rate-efficient estimator and show the accuracy of the latter in comparison with a rate-optimal estimator based on the pre-estimation of the spot volatility. Finally, using the rate-optimal Fourier estimator, we reconstruct the series of the daily volatility of volatility of the S&P500 and EUROSTOXX50 indices over long samples and provide novel insight into the existence of stylized facts about the volatility of volatility dynamics.

Suggested Citation

  • Giacomo Toscano & Giulia Livieri & Maria Elvira & Stefano Marmi, 2024. "Volatility of Volatility Estimation: Central Limit Theorems for the Fourier Transform Estimator and Empirical Study of the Daily Time Series Stylized Facts," Journal of Financial Econometrics, Oxford University Press, vol. 22(1), pages 252-296.
    • Handle: RePEc:oup:jfinec:v:22:y:2024:i:1:p:252-296.
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    File URL: http://hdl.handle.net/10.1093/jjfinec/nbac035
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    Keywords

    central limit theorem; Fourier analysis; non-parametric estimation; stochastic volatility; volatility of volatility;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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